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Theorem prnz 4686
 Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
Hypothesis
Ref Expression
prnz.1 𝐴 ∈ V
Assertion
Ref Expression
prnz {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz
StepHypRef Expression
1 prnz.1 . . 3 𝐴 ∈ V
21prid1 4672 . 2 𝐴 ∈ {𝐴, 𝐵}
32ne0ii 4275 1 {𝐴, 𝐵} ≠ ∅
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2114   ≠ wne 3011  Vcvv 3469  ∅c0 4265  {cpr 4541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ne 3012  df-v 3471  df-dif 3911  df-un 3913  df-nul 4266  df-sn 4540  df-pr 4542 This theorem is referenced by:  opnz  5342  propssopi  5375  fiint  8783  wilthlem2  25652  upgrbi  26884  wlkvtxiedg  27412  shincli  29143  chincli  29241  spr0nelg  43933  sprvalpwn0  43940
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